1. INTRODUCTION

1.1. Historical Context of the Time Problem

Modern theoretical physics is in a state of conceptual crisis caused by the fundamental incompatibility of time interpretations in two fundamental theories:

• General Relativity (GR): Time is a dynamic coordinate of a four-dimensional manifold, covariant with space and dependent on the metric tensor $g_{\mu\nu}$. In GR, time is not absolute but depends on the gravitational field and the observer's motion. The time interval $d\tau$ between two events is determined by the metric:

• Quantum Mechanics (QM): Time acts as an external absolute parameter $t$ governing the unitary evolution of the wave function $\Psi(t)$ through the Schrödinger equation:

where $\hat{H}$ is the system's Hamiltonian. In QM, time is a classical parameter that is not quantized.

1.2. The Wheeler-DeWitt Problem

Attempting to unify these approaches within canonical quantum gravity leads to the Wheeler-DeWitt equation [1]:

where $\Psi[g_{ij}, \phi]$ is the wave function of the universe, depending on the three-dimensional metric $g_{ij}$ and matter fields $\phi$. This equation states that the total energy of a closed universe is zero, and therefore the quantum state of the universe is stationary:

This is a fundamental contradiction: the equation does not explicitly contain time, which contradicts the observed dynamics of cosmological expansion, galaxy evolution, and the entire observable universe.

1.3. Approaches to Solving the Problem

Many approaches have been proposed to resolve this problem:

1. Time Gauge: Introducing external time through gauge choice, which violates general covariance.
2. Many-worlds interpretation: Different "moments of time" exist as parallel universes, but this does not explain the perception of time flow.
3. Emergent time: Time arises from a more fundamental timeless layer, but specific mechanisms remained unclear.

1.4. Our Approach: Theory of Recursive Emergence

In this work, we propose the Theory of Recursive Emergence. We assert that the static nature of the quantum state and the dynamics of the classical world do not contradict each other, but belong to different phase states of matter, separated by a critical complexity threshold.

Key idea: time is not a fundamental quantity, but arises as an order parameter as a result of a second-order phase transition when the quantum complexity of the system exceeds a critical value.

2. FORMALISM OF INFORMATION LAYER $\mathcal{U}$

2.1. Ontology of the Timeless Layer

The fundamental ontological element of reality in our theory is the "Timeless Information Layer", denoted $\mathcal{U}_n$, where the index $n$ indicates the level of the recursive hierarchy. Mathematically, it is defined by a triple:

Where:

• $\mathcal{H}_{tot} = \bigotimes_i \mathcal{H}_i$ — the full Hilbert space of the system, representing the tensor product of local Hilbert spaces $\mathcal{H}_i$ (a network of qubits or more general quantum systems). The dimension $\dim(\mathcal{H}_{tot}) = \prod_i \dim(\mathcal{H}_i)$ grows exponentially with the number of subsystems.
• $\mathcal{A}$ — $C^*$-algebra of local observables, closed under addition, multiplication, conjugation, and topological closure operations. Elements $\hat{A} \in \mathcal{A}$ represent local measurable quantities accessible to an observer in a given region of space.
• $\rho_n$ — density matrix of a pure state, satisfying the condition:

where $\hat{H}_{fund}$ is the fundamental Hamiltonian of layer $\mathcal{U}_n$. This condition means that state $\rho_n$ is stationary with respect to fundamental dynamics.

2.2. Absence of Space-Time

In layer $\mathcal{U}_n$, there is no metric space or time in their classical understanding. Instead:

1. Topology is given by the quantum entanglement graph: Graph vertices correspond to local subsystems, and edges correspond to entangled states between them. The entanglement graph $G = (V, E)$ defines the structure of connections in the system.
2. Spatial proximity is determined through Mutual Information: For two subsystems $A$ and $B$, mutual information is defined as:

where $S(\rho) = -\text{Tr}(\rho \ln \rho)$ is the von Neumann entropy, $\rho_A = \text{Tr}_B(\rho_{AB})$ is the reduced density matrix of subsystem $A$.

We postulate that the "distance" between subsystems is inversely proportional to their mutual information:

where $\alpha > 0$ is a constant with dimension of length, and $\epsilon > 0$ is a regularization parameter preventing divergence when $I(A:B) \to 0$.

This corresponds to the ER=EPR hypothesis [4], asserting the equivalence of entanglement and Einstein-Rosen bridges: entangled particles are connected by wormholes in space-time.

2.3. Recursive Hierarchy of Layers

A key feature of our theory is the recursive structure of information layers:

Each layer $\mathcal{U}_{n+1}$ arises from the previous $\mathcal{U}_n$ when local critical complexity is reached. This creates a hierarchical structure where each level has its own "physics" and laws.

3. QUANTUM COMPLEXITY AS EVOLUTION PARAMETER

3.1. Definition of Quantum Complexity

In the absence of time $t$, the evolution of the system between layers is described by changes in information structure. We postulate that the true evolution parameter is Quantum Complexity ($\mathcal{C}$) [2].

For a pure state $|\Psi\rangle$, quantum complexity is defined as the minimum number of elementary unitary operations (gates) required to synthesize the current state $|\Psi\rangle$ from a reference state $|\Psi_0\rangle$:

where $\mathcal{G}$ is the set of all possible unitary operations constructed from a set of elementary gates.

For a mixed state $\rho$, complexity is defined through purification:

3.2. Properties of Quantum Complexity

Quantum complexity has the following key properties:

1. Monotonicity: For unitary evolution $U(t)$, complexity monotonically increases:
   $$\frac{d\mathcal{C}}{dt} \geq 0$$
   This follows from the second law of thermodynamics for quantum systems.
2. Scaling: For a system of $N$ qubits, complexity grows exponentially:
   $$\mathcal{C}_{\max} \sim 2^N$$
3. Critical value: There exists a critical complexity $\mathcal{C}_{crit}$ at which a phase transition occurs:
   $$\mathcal{C}_{crit} = \alpha N \ln N$$
   where $\alpha$ is a dimensionless constant of order unity.

3.3. Connection with Entropy and Entanglement

Quantum complexity is closely related to entanglement entropy, but not identical to it. While entropy measures the amount of information, complexity measures the "structuredness" of that information.

For a system with high entanglement:

where $S$ is the von Neumann entropy.

4. DYNAMICS OF TIME EMERGENCE

4.1. Time Field as Order Parameter

We introduce the time field $\mathcal{T}(x, \mathcal{C})$ as a scalar order parameter, depending on spatial coordinates $x$ (defined through the information metric) and quantum complexity $\mathcal{C}$.

The dynamics of time emergence is described as a second-order phase transition upon reaching critical complexity $\mathcal{C}_{crit}$. This is analogous to Bose-Einstein condensation or the superconductor-normal metal transition.

4.2. Seledchik Equation: Derivation and Justification

We postulate the following differential renormalization group flow equation:

Where:

• $\mu > 0$ — relaxation constant, determining the rate of time growth in the supercritical phase. Physically, $\mu$ is related to the characteristic relaxation time of the system.
• $\xi > 0$ — saturation constant, preventing unlimited growth of $\mathcal{T}$. The term $-\xi \mathcal{T}^3$ provides nonlinear saturation.
• $\eta > 0$ — diffusion coefficient, describing spatial alignment of the time field. The term $\eta \nabla^2 \mathcal{T}$ ensures local consistency of the temporal coordinate.
• $\nabla^2$ — Laplacian, defined through the information metric of layer $\mathcal{U}$.

4.3. Justification of Equation Form

The equation has the standard form of the Landau-Ginzburg equation for a second-order phase transition:

1. Linear term $\mu(1 - \mathcal{C}_{crit}/\mathcal{C})\mathcal{T}$ describes the growth of the order parameter in the supercritical phase. The coefficient changes sign at $\mathcal{C} = \mathcal{C}_{crit}$.
2. Cubic term $-\xi \mathcal{T}^3$ provides stabilization and saturation. Without it, the order parameter would grow unboundedly.
3. Diffusion term $\eta \nabla^2 \mathcal{T}$ ensures spatial coherence, preventing the formation of domains with different time values.

4.4. Analysis of Equation Solutions

Subcritical Phase ($\mathcal{C} < \mathcal{C}_{crit}$)

In the subcritical phase, the coefficient of the linear term is negative:

The only stable solution is trivial:

This corresponds to the state of layer $\mathcal{U}$ without macroscopic time. The system is in a purely quantum timeless state.

Critical Point ($\mathcal{C} = \mathcal{C}_{crit}$)

At the critical point, the coefficient becomes zero, and the equation becomes:

This is a bifurcation point where the trivial solution loses stability.

Supercritical Phase ($\mathcal{C} > \mathcal{C}_{crit}$)

In the supercritical phase, a pitchfork bifurcation occurs. The trivial solution $\mathcal{T} = 0$ loses stability, and two new stable solutions arise:

Near the critical point ($\mathcal{C} \gtrsim \mathcal{C}_{crit}$), solutions behave as:

This is the classical behavior of an order parameter in a second-order phase transition with critical exponent $\beta = 1/2$.

4.5. Critical Exponents and Universality

Analysis of the equation near the critical point allows determination of critical exponents:

• Exponent $\beta$: Determines the behavior of the order parameter:
  $$\mathcal{T} \sim (\mathcal{C} - \mathcal{C}_{crit})^\beta, \quad \beta = \frac{1}{2}$$
• Exponent $\nu$: Determines the correlation length:
  $$\xi_{corr} \sim (\mathcal{C} - \mathcal{C}_{crit})^{-\nu}$$
  From the diffusion term, it follows that $\nu = 1/2$.
• Exponent $\gamma$: Determines susceptibility:
  $$\chi = \frac{\partial \mathcal{T}}{\partial h} \sim (\mathcal{C} - \mathcal{C}_{crit})^{-\gamma}, \quad \gamma = 1$$
  where $h$ is an external field breaking the symmetry.

These exponents correspond to the mean-field universality class, indicating that the phase transition has a global character.

5. SPONTANEOUS CPT-SYMMETRY BREAKING

5.1. Interpretation of Two Solutions

The presence of two solutions $\pm \mathcal{T}$ is interpreted as a physical splitting of reality into two causally unconnected branches:

• Branch $t_+$ ($\mathcal{T} > 0$): Our Universe with matter dominance, forward entropy growth, and standard thermodynamic arrow of time.
• Branch $t_-$ ($\mathcal{T} < 0$): CPT-conjugate Universe with antimatter dominance, reverse time flow relative to $t_+$, and reverse thermodynamic arrow.

5.2. CPT Theorem and Its Violation

The CPT theorem states that any local quantum field theory is invariant under the combined operation of charge conjugation (C), parity (P), and time reversal (T). However, in our theory, spontaneous violation of this symmetry occurs.

At the moment of phase transition ($\mathcal{C} = \mathcal{C}_{crit}$), the system randomly chooses one of the two branches (or depending on fluctuations). This symmetry breaking is analogous to symmetry breaking in the theory of spontaneous magnetization.

5.3. Solution to the Baryon Asymmetry Problem

The classical baryon asymmetry problem is that the observable universe is dominated by matter over antimatter, although in the early universe they should have been present in equal amounts.

Our theory offers an elegant solution: at the moment of the "Big Bang" (phase transition $\mathcal{C} = \mathcal{C}_{crit}$), matter and antimatter were separated into different temporal flows:

• Matter → branch $t_+$
• Antimatter → branch $t_-$

Global balance is preserved, but locally (in each branch) asymmetry is observed. This explains why we do not observe antimatter in our universe — it is in the causally isolated branch $t_-$.

5.4. Causal Isolation of Branches

The two branches $t_+$ and $t_-$ are causally isolated because:

1. They exist in different information layers after the phase transition.
2. Communication between them would require local reduction of complexity below the critical value, which is thermodynamically forbidden.
3. Any attempt at communication between branches would violate the second law of thermodynamics.

6. DERIVATION OF GRAVITY FROM INFORMATION

6.1. Information Geometry

We show that the classical metric $g_{\mu\nu}$ is an effective macroscopic description of the microphysics of layer $\mathcal{U}$.

Fisher-Bures Metric

The distance between quantum states on the parameter manifold is given by the Fisher-Bures metric. For a pure state $|\Psi(\theta)\rangle$ depending on parameters $\theta = \{\theta^\mu\}$, the information metric is defined as:

where $\partial_\mu = \partial/\partial \theta^\mu$.

This metric has the following properties:

• Positive definiteness: $G_{\mu\nu} v^\mu v^\nu \geq 0$ for any vector $v$.
• Invariance under phase transformations: $|\Psi\rangle \to e^{i\phi}|\Psi\rangle$ does not change the metric.
• Monotonicity: The metric does not increase under quantum operations.

Identification with Space-Time

We postulate that space-time arises from information geometry:

where $\ell_P = \sqrt{\hbar G/c^3}$ is the Planck length, ensuring correct dimensionality, and $\theta(x)$ are parameters depending on spatial coordinates $x$.

This identification means that space-time geometry encodes information about quantum states at each point.

6.2. Thermodynamics of Event Horizon

Area Law for Entropy

Using T. Jacobson's approach [3], consider a local causal horizon. The entanglement entropy $S$ of the horizon obeys the Area Law:

where $A$ is the horizon area in Planck units.

This law follows from the fact that entanglement entropy between the interior and exterior regions of the horizon is proportional to the area of their boundary.

Unruh Temperature

For an accelerated observer with acceleration $a$, there exists an effective temperature (Unruh temperature):

For an event horizon with surface gravity $\kappa$, the temperature is:

First Law of Thermodynamics

When energy flow $\delta Q$ passes through the horizon, the first law of thermodynamics holds:

Substituting expressions for temperature and entropy:

6.3. Derivation of Einstein's Equations

Raychaudhuri Equation

The change in horizon area is related to the energy-momentum tensor through the Raychaudhuri equation. For a causal horizon:

where $\theta$ is the expansion of the congruence, and $\lambda$ is an affine parameter.

The Raychaudhuri equation relates expansion to the Ricci tensor:

where $k^\mu$ is the tangent vector to the horizon, $\sigma_{\mu\nu}$ is the shear tensor.

Thermodynamic Identity

Combining the thermodynamic relation with the geometric expression for area change, we obtain:

Using locality and covariance, this leads to the exact relation:

Thus, Einstein's equations are derived as the equation of state for equilibrium entropy of the information layer.

Interpretation of Cosmological Constant

The cosmological constant $\Lambda$ is interpreted as the residual information density of the vacuum:

where $\rho_{info}^{vac}$ is the information density of the vacuum state of layer $\mathcal{U}$.

7. RESOLUTION OF PHYSICAL PARADOXES

7.1. Cosmological Singularity Problem

Classical Singularity

In classical GR, the point $t=0$ (Big Bang) is a singularity where:

• Energy density $\rho \to \infty$
• Space-time curvature $R \to \infty$
• The metric becomes degenerate

This indicates incompleteness of the theory in this region.

Our Solution

In our theory, the point $t=0$ is not a singularity, but represents a phase transition point $\mathcal{C} = \mathcal{C}_{crit}$, where:

1. Classical time $\mathcal{T}$ decays to zero: $\mathcal{T} \to 0$ as $\mathcal{C} \to \mathcal{C}_{crit}$.
2. Physics transitions to a regime of pure quantum information without classical space-time.
3. Density and curvature remain finite, as they are determined by the information structure of layer $\mathcal{U}$, not the classical metric.
4. "Before" the Big Bang, the system existed in a timeless layer $\mathcal{U}_0$ with high quantum complexity, but without macroscopic time.

Recursive Structure

A recursive structure is possible, where "before" the Big Bang there existed a previous layer $\mathcal{U}_{-1}$, which also underwent a phase transition, giving rise to our layer $\mathcal{U}_0$. This eliminates the problem of the "beginning" of the universe.

7.2. Black Hole Information Paradox

Classical Paradox

The black hole information paradox is formulated as follows:

• Quantum mechanics requires unitary evolution: information cannot be destroyed.
• Classical GR predicts that information falling into a black hole is lost behind the event horizon.
• Hawking radiation is thermal and does not carry information about the initial state.

This creates a contradiction between unitarity and information loss.

Our Solution

In our theory, black holes are regions of space where local quantum complexity reaches the saturation limit:

This leads to:

1. Local disappearance of time: In the black hole region, the time field $\mathcal{T} \to 0$, as complexity reaches its maximum.
2. Formation of transition to new layer: The black hole becomes a "portal" to the next recursive layer $\mathcal{U}_{n+1}$.
3. Information preservation: Information is not destroyed, but encoded in the structure of the next layer $\mathcal{U}_{n+1}$, preserving unitarity at the global level.
4. Absence of singularity: The center of the black hole is not a singularity, but a phase transition point between layers.

Connection with AdS/CFT Hypothesis

Our model is consistent with the AdS/CFT correspondence hypothesis [4], where gravity in the bulk is equivalent to a conformal field theory on the boundary. In our interpretation, a black hole is a transition between different descriptions of the same information structure.

7.3. Arrow of Time

Irreversibility Problem

The classical problem of the arrow of time is that fundamental laws of physics (quantum mechanics, GR) are time-reversible, but the observed world demonstrates clear irreversibility (entropy growth, aging, decay).

Our Explanation

In our theory, thermodynamic irreversibility of time is a consequence of global growth of Quantum Complexity:

We perceive the flow of time because:

1. The system's complexity continuously increases: $\partial \ln \mathcal{C} / \partial t > 0$.
2. This increase in complexity is irreversible due to the second law of thermodynamics for quantum systems.
3. The time parameter $\mathcal{T}$ is related to complexity through the Seledchik equation, so complexity growth leads to the perception of time flow.
4. Time reversal would require a decrease in complexity, which is thermodynamically forbidden.

Connection with Entropy

Complexity growth is closely related to entropy growth:

This explains why the arrow of time coincides with the arrow of entropy.

8. CONNECTION WITH OTHER THEORIES

8.1. Connection with Loop Quantum Gravity

Our theory has points of contact with Loop Quantum Gravity (LQG):

• Discrete structure: In LQG, space-time is discrete at the Planck scale. In our theory, this corresponds to the discrete structure of information layer $\mathcal{U}$.
• Spin network: In LQG, the state is described by a spin network. In our theory, this corresponds to the entanglement graph in layer $\mathcal{U}$.
• Absence of singularities: LQG predicts the absence of Big Bang singularities. Our theory also eliminates singularities through phase transitions.

However, a key difference: in LQG, time remains fundamental, while in our theory it is emergent.

8.2. Connection with String Theory

String theory also offers a solution to the time problem through compactification of extra dimensions. In our theory:

• Extra dimensions can be interpreted as internal degrees of freedom of information layer $\mathcal{U}$.
• Compactification corresponds to the transition from a high-dimensional layer $\mathcal{U}_n$ to a low-dimensional effective description.
• Dualities in string theory may correspond to different representations of the same information layer.

8.3. Connection with Causal Dynamical Triangulation

Causal Dynamical Triangulation (CDT) constructs space-time from elementary simplices. In our theory:

• Elementary simplices correspond to local regions of information layer $\mathcal{U}$.
• Causal structure arises from the causal structure of the entanglement graph.
• Dynamics is determined by quantum complexity growth, not by summing over histories.

9. EXPERIMENTAL CONSEQUENCES AND PREDICTIONS

9.1. CPT-Symmetry Violation

Our theory predicts spontaneous CPT-symmetry violation on cosmological scales. This may manifest in:

• Asymmetry in the distribution of matter and antimatter (which is already observed).
• Violation of time reversibility in cosmological processes.
• Anomalies in cosmic microwave background radiation related to asymmetry between different regions of the sky.

9.2. Gravity Modifications on Small Scales

At Planck scales ($\ell_P \sim 10^{-35}$ m), our theory predicts deviations from classical GR:

where the correction $\delta g_{\mu\nu}$ depends on local quantum complexity.

9.3. Quantum Correlations in Cosmic Microwave Background

The entanglement graph in layer $\mathcal{U}$ should leave traces in the large-scale structure of the universe and cosmic microwave background. This may manifest in:

• Anomalous correlations in the angular power spectrum of CMB.
• Large-scale anomalies in galaxy distribution.
• Violations of statistical isotropy.

9.4. Black Hole Behavior

Our theory predicts that black holes do not have singularities at their centers, but represent transitions between information layers. This may manifest in:

• Absence of true singularities in solutions of Einstein's equations.
• Modification of Hawking radiation spectrum at late stages of evaporation.
• Possibility of information "tunneling" from a black hole through transition to another layer.

10. NUMERICAL ESTIMATES AND SCALES

10.1. Critical Complexity

For the observable universe with $N \sim 10^{80}$ baryons, the estimate of critical complexity:

This corresponds to the complexity necessary to describe the state of the universe at the moment of the Big Bang.

10.2. Characteristic Time of Phase Transition

The relaxation time $\tau$ of the phase transition is related to constant $\mu$:

This is Planck time, which is consistent with the scale at which quantum gravity should occur.

10.3. Scale of CPT Violation

CPT-symmetry violation should be most noticeable on cosmological scales ($\sim 10^{26}$ m) and time scales of the age of the universe ($\sim 10^{17}$ s).

11. CONCLUSION

The Theory of Recursive Emergence offers a unified conceptual framework combining quantum information theory, thermodynamics, and gravity. Key achievements of the theory:

11.1. Main Results

1. Time condensation equation: The Seledchik equation is formulated, describing the dynamics of time emergence as a second-order phase transition.
2. CPT violation: The nature of spontaneous CPT-symmetry breaking is explained through time bifurcation and splitting of reality into two causally isolated branches.
3. Derivation of gravity: Derivation of Einstein's equations from thermodynamics of the information layer is demonstrated, showing that gravity is an emergent phenomenon.
4. Elimination of singularities: Cosmological singularities (Big Bang, black hole centers) are replaced by phase transitions in a recursive hierarchy of information layers.
5. Explanation of arrow of time: Thermodynamic irreversibility is explained by global growth of quantum complexity.

11.2. Philosophical Consequences

The theory has profound philosophical consequences:

• Ontology of time: Time is not a fundamental entity, but arises from a more basic information structure.
• Multiple universes: The existence of CPT-conjugate branch $t_-$ means the existence of a "parallel" universe with reverse arrow of time.
• Recursiveness of reality: The possibility of an infinite hierarchy of information layers raises the question of the "beginning" and "end" of reality.
• Connection between information and matter: Matter and space-time are manifestations of information structure, not vice versa.

11.3. Directions for Further Research

Further development of the theory involves:

1. Numerical calculations: Computing critical exponents $\mu$, $\xi$, $\eta$ within lattice models of quantum gravity.
2. Quantum complexity: Development of efficient algorithms for computing quantum complexity for large systems.
3. Cosmological applications: Application of the theory to specific cosmological models and comparison with observational data.
4. Experimental verification: Search for experimental signatures of CPT-symmetry violation and gravity modifications.
5. Connection with other approaches: Establishing closer connections with loop quantum gravity, string theory, and other approaches to quantum gravity.

The Theory of Recursive Emergence opens new perspectives for understanding the fundamental nature of time, space, and information in the universe.